Fractions PDF

CHAPTER 2 FRACTIONS I believe five out of four people have trouble with fractions. --Steven Wright Numerators, Denominators, and Reciprocals of Fractio...

CHAPTER 2 FRACTIONS I believe five out of four people have trouble with fractions. --Steven Wright Numerators, Denominators, and Reciprocals of Fractions A fraction indicates a portion of a whole number. There are differnt types of fractions: common or proper fractions such as ½ or ¾ improper fractions such as 4/2 or 3/1 mixed fraction or mixed number such as 1 ½ or 21 ¾ complex fraction such as 1 ½ or 3 ¾ 2¾ 4 5 /6 decimal fractions such as 0.5 or 0.75 (we already covered these in Chapter 1). A fraction is an expression of division, with one number placed over another number. The bottom number, or denominator, indicates the total number of parts into which the whole is divided. The top number, or numerator, indicates how many of those parts are considered. The fraction may be read as the “numerator divided by the denominator.” Numerator Denominator Examples ¼ = 1 part of 4 parts, or ¼ of the whole = 0.25 Pharmaceutial Calculations FRACTIONS 1 GPL v3.0 or later, 2012 by Sean Parsons  ½ = 1 part of 2 parts, or ½ of the whole = 0.5 Reciprocals of fractions To find the reciprocal of a fraction simply switch the numerator and denominator (flip it over). 2 3 The reciprocal of3is 2 4 5 The reciprocal of5is4 A whole number could be considered to have a denominator of one, so the reciprocal of a whole number would be 1 over the original whole number. 1 The reciprocal of 5 is5 1 The reciprocal of 9 is9 The following page is a worksheet to help reinforce this information. Pharmaceutial Calculations FRACTIONS 2 GPL v3.0 or later, 2012 by Sean Parsons  Practice 2.1 Convert the following fractions to decimals. 1 2 2 1) 5 = _____ 8) = _____ 15) 22 = _____ 3 1 3 5 2) 2 = _____ 9) 4 = _____ 16) = _____ 35 2 4 9 3) 4 = _____ 10) 5 = _____ 17) = _____ 55 1 3 13 4) 10 = _____ 11) 8 = _____ 18) = _____ 63 1 5 15 5) 12 = _____ 12) = _____ 19) = _____ 12 71 1 7 23 6) 100 = _____ 13) = _____ 20) = _____ 18 83 1 4 7) 1000 = _____ 14) = _____ 11 Determine the reciprocal of the following fractions. 1 2 2 21) 5 = _____ 28) = _____ 35) 22 = _____ 3 1 3 5 22) 2 = _____ 29) 4 = _____ 36) = _____ 35 2 4 9 23) 4 = _____ 30) = _____ 37) = _____ 5 55 1 3 13 24) 10 = _____ 31) = _____ 38) = _____ 8 63 1 5 15 25) 12 = _____ 32) = _____ 39) = _____ 12 71 1 7 23 26) 100 = _____ 33) = _____ 40) = _____ 18 83 1 4 27) 1000 = _____ 34) = _____ 11 Pharmaceutial Calculations FRACTIONS 3 GPL v3.0 or later, 2012 by Sean Parsons Reducing fractions to lowest terms In reducing a fraction to lowest terms (also called simplifying), you need to divide the numerator and denominator by their greatest common factor. When answering pharmacy related math questions, fractions should always be reduced to lowed terms. The greatest common factor is the largest whole number that can divide into the numerator and the denominator. Some fractions are already in lowest terms if there is no factor common to the numerator and the denominator. The steps to simplify a fraction: List the whole number factors of the numerator and the denominator. Find the factors common to both the numerator and denominator. Divide the numerator and denominator by the largest common factor. 30 Example 70 List the factors of the numerator and the denominator: Numerator: 1,2,3,5,6,10,15,30 Denominator: 1,2,5,7,10,14,35,70 The greatest common factor is 10 Divide the numerator and the denominator by 10 Numerator: 30 ÷ 10 = 3 Denominator: 70 ÷ 10 = 7 3 is the fraction when reduced to lowest terms. 7 Practice Problems Reduce the fractions in the following problems. 4 1)8 55 2)99 36 3) 144 Pharmaceutial Calculations FRACTIONS 4 GPL v3.0 or later, 2012 by Sean Parsons  Practice 2.2 Reduce the following fractions to lowest terms. 2 12 1) 4 = _____ 16) 35 = _____ 3 125 2) 9 = _____ 17) = _____ 500 2 270 3) 16 = _____ 18) = _____ 2700 18 65 4) 54 = _____ 19) = _____ 585 20 82 5) 240 = _____ 20) = _____ 164 25 79 6) 125 = _____ 21) = _____ 237 55 17 7) 75 = _____ 22) = _____ 102 28 19 8) 35 = _____ 23) = _____ 285 63 18 9) 90 = _____ 24) 81 = _____ 15 121 10) 75 = _____ 25) 605 = _____ 14 63 11) = _____ 26) 135 = _____ 36 7 42 12) 11 = _____ 27) 72 = _____ 0 33 13) 18 = _____ 28) 77 = _____ 34 108 14) 51 = _____ 29) 180 = _____ 14 110 15) 63 = _____ 30) 363 = _____ Pharmaceutial Calculations FRACTIONS 5 GPL v3.0 or later, 2012 by Sean Parsons Adding and subtracting fractions In order to add and subtract fractions, they must have a common denominator. Once they have a common denominator, you may add and subtract the numerators like you would in an ordinary addition or subtraction problem and then write the sum or the difference over top of the common denominator. How to create equivalent fractions with a common denominator: Find a multiple for the denominator of both numbers. Rewrite the fractions as equivalent fractions with the common denominator. 4 1 Example 5 −3 You can multiply the denominators to find a common denominator: 5 × 3 = 15 Now create equivalent fractions: 4 3 12 1 5 5 5 × 3 = 15 3 × 5 = 15 Now let's solve the problem with the equivalent fractions: 12 5 7 15 − 15 = 15 Practice Problems Solve the following practice problems. 1 1 1) 3 −5 3 3 2) 8 +5 1 3 5 7 3) 19 + 38 + 76 + 152 Pharmaceutial Calculations FRACTIONS 6 GPL v3.0 or later, 2012 by Sean Parsons  Practice 2.3 Solve the following problems. 1 1 7 3 1) 3 +3 = _____ 13) 5 + 10 + 1000 = _____ 5 3 5 4 1 2) 8 −8 = _____ 14) + 27 + 48 = _____ 8 7 3 7 3 3) 8 −8 = _____ 15) 1 + 100 + 10 = _____ 4 1 1 1 1 4) 25 +5 = _____ 16) 2 + 9 + 36 = _____ 4 1 1 1 1 5) 25 +5 = _____ 17) 3 +6+9 = _____ 7 1 11 4 11 6) 8 −4 = _____ 18) 99 + 9 + 33 = _____ 2 3 1 1 1 1 3 7) 7 + 7 + 7= _____ 19) 8 +4+2+8 = _____ 1 2 7 12 14 1 1 8) 8 + 8 + 8= _____ 20) 19 + 38 + 19 + 38 = _____ 14 1 1 2 7 9) 38 − 19 = _____ 21) +8+8 = _____ 8 11 2 10 13 1 17 10) 12 −3 = _____ 22) 30 + 15 + 5 + 60 = _____ 9 1 1 2 7 11) 24 −6 = _____ 23) +8+8 = _____ 8 4 67 1 2 1 12) 5 − 100 = _____ 24) + 27 + 3 = _____ 81 Pharmaceutial Calculations FRACTIONS 7 GPL v3.0 or later, 2012 by Sean Parsons Multiplying fractions Reduce if possible. Multiply the numerators of the fractions to get the new numerator. Multiply the denominators of the fractions to get the new denominator. Simplify the resulting fraction if possible. 2 4 Example 3 × 7= ? Reduce if possible 2 4 3 × 7no reduction is possible Multiply the numerators 2×4= 8 Multiply the denominators 3 × 7 = 21 Simplify the resulting fraction if possible 8 21 is already in simplest terms 2 4 8 3 × 7 = 21 Practice Problems Multiply the following fractions. 2 1 1) 5 × 8= 2 2 2) 3 × 3= 7 9 3) 8 × 21= Pharmaceutial Calculations FRACTIONS 8 GPL v3.0 or later, 2012 by Sean Parsons Multiplying mixed numbers Write mixed numbers as improper fractions. Reduce if possible. Multiply the numerators of the fractions to get the new numerator. Multiply the denominators of the fractions to get the new denominator. Simplify the resulting fraction if possible. 1 5 Example 4 8 × 1 11= ? Write mixed numbers as improper fractions. 33 16 8 × 11 Reduce if possible 3 2 33 16 × 18 111 Multiply the numerators 3×2= 6 Multiply the denominators 1×1= 1 Simplify the resulting fraction if possible 6 1 =6 3 2 1 5 33 16 6 4 8 × 1 11 = × =1=6 18 111 Practice Problems Solve the following practice problems by multiplying the fractions. 1 4 1) 13 3 × 1 5= 1 3 2) 7 5 × 3 4= 2 4 3) 20 3 × 1 31= Pharmaceutial Calculations FRACTIONS 9 GPL v3.0 or later, 2012 by Sean Parsons  Practice 2.4 Solve the following problems. 1 1 0 5 1) 3 of3 = _____ 7) of6 = _____ 4 1 3 1 3 2) 8 of8 = _____ 8) of5 = _____ 2 2 1 1 1 3) 3 of3 = _____ 9) of2 = _____ 9 5 3 2 1 4) of8 = _____ 10) 7 of5 = _____ 8 1 3 9 3 5) of 5 5 = _____ 11) of = _____ 10 4 7 3 3 3 6) 8 of8 = _____ 12) 11 of4 = _____ Multiply the following fractions. 3 1 6 13) 10 × 4 = _____ 17) × 21 = _____ 7 13 7 3 14) 15 × 8 = _____ 18) 25 × 10 = _____ 21 2 8 5 15) 25 × 5 = _____ 19) × 6 = _____ 15 49 1 11 2 16) 50 × 4 = _____ 20) × 33 = _____ 14 Multiply the following mixed numbers. 4 1 21) 45 × 16 = 1 2 22) 72 × 23 = 17 1 23) 1 18 × 1 5 = 5 1 24) 1 7 × 10 2 = Pharmaceutial Calculations FRACTIONS 10 GPL v3.0 or later, 2012 by Sean Parsons Dividing Fractions Dividing fractions is just like multiplying fractions (with one exception), you need to “flip” (find the reciprocal of) the fraction you are dividing by. The following are the steps to dividing fractions: Find the reciprocal of the fraction you are dividing by. Reduce if possible. Multiply the numerators of the fractions to get the new numerator. Multiply the denominators of the fractions to get the new denominator. Simplify the resulting fraction if possible. A simpler way of remembering this is to simply think “Flip and Multiply” 2 4 Example 3 ÷7 “Flip and Multiply” 1 2 7 7 1 × = 6 = 16 3 24 Practice Problems Solve the following problems. 2 1 1) 5 ÷ 8= 2 2 2) 3 ÷ 3= 7 9 3) 8 ÷ 21= Pharmaceutial Calculations FRACTIONS 11 GPL v3.0 or later, 2012 by Sean Parsons  Practice 2.5 Divide the following fractions: 1 1 9 9 1) ÷ 3= 8) 10 ÷ 10= 3 1 3 7 5 2) 8 ÷ 8= 9) ÷ 7= 5 1 3 3 3) 8 ÷ 16= 10) 12 ÷ 8= 4 1 2 1 4) 25 ÷ 5= 11) 3 3 ÷ 1 4= 2 1 5 1 5) 3 ÷ 2= 12) 1 6 ÷ 7 3= 16 8 3 1 6) 27 ÷ 9= 13) 15 4 ÷ 5 7= 5 1 7) 15 ÷ 5= 14) 15 ÷ 1 5= Solve the following word problems: 15) How many 6 ¼ milligram capsules of metoprolol tartrate can be made from 500 mg of metoprolol tartrate? 16) The OR likes to use phenylephrine syringes, each containing 4/5 of a milligram. How many phenylephrine syringes can I make if I have a 20 mg vial on hand? Pharmaceutial Calculations FRACTIONS 12 GPL v3.0 or later, 2012 by Sean Parsons  Practice 2.6 Convert these fractions into equivalent decimals (you only need to solve them out to the thousandths position). 7 13 9 1) 15 = 4) 15 = 7) = 27 13 17 17 2) 40 = 5) = 8) = 30 25 5 7 13 3) = 6) 8 = 9) 37 = 8 Determine the reciprocals of the following fractions. 3 15 65 10) 4 = 12) = 14) = 24 585 1 2 11) = 13) = 15) 1 = 5 8 Reduce the following fractions. 6 2 27 16) 8 = 19) 8 = 22) 45 = 12 12 17 17) 48 = 20) 36 = 23) = 31 15 18 52 18) = 21) 24 = 24) = 24 148 Perform the following additions and subtractions. 1 3 1 2 25) 6 +6 = 29) 8 +3 = 7 5 11 1 26) + 24 = 30) 16 −2 = 6 7 7 1 4 27) − 12 = 31) 6 +7 = 8 5 2 9 2 28) −3 = 32) 11 −5 = 6 Perform the following multiplications and divisions. 1 4 7 33) 6 ×7 = 36) 16 ÷7 = 7 1 34) 4×8 = 37) 10 ÷ 2 = 3 1 7 35) 4 ÷6 = 38) 16 ×7 = Pharmaceutial Calculations FRACTIONS 13 GPL v3.0 or later, 2012 by Sean Parsons  3 1 3 1 39) 4 ×3 = 40) 4 ÷3 = Solve the following word problems. 41) What is the total volume of a mixture of 1/8 milliliter of solution X mixed with 2 and 1/4 milliliters of solution Y? 42) A patient is to receive 3 liters of an intravenous solution. If 2 1/8 liters have already been administered, how much solution remains? 43) How many milligrams of drug are needed to make 20 tablets of 1/4 milligrams each? 44) A bottle of Children's Tylenol contains 20 teaspoons of liquid. If each dose for a 2 year-old child is 1/2 teaspoon, how many doses are available in this bottle? 45) A patient is on strict recording of fluid intake and output, including measurements of liquid medications. A nursing student gave the patient 1/4 ounce of medication at 8 AM and 1/3 ounce of medication at 12 noon. What is the total amount of medication to be recorded on the Intake and Output sheet? Pharmaceutial Calculations FRACTIONS 14 GPL v3.0 or later, 2012 by Sean Parsons